Solving bar brackets, also known as absolute value brackets, involves understanding the concept of absolute value. The absolute value of a number is its distance from zero on the number line, regardless of direction. Here’s how you can solve equations involving bar brackets:
Step 1: Understand the Absolute Value
The absolute value of a number is always non-negative. For example, |3| = 3 and |-3| = 3. This means that the absolute value of both 3 and -3 is 3.
Step 2: Set Up the Equation
When you encounter an equation with absolute value brackets, such as |x| = 5, you need to consider two scenarios:
- x = 5
- x = -5
Step 3: Solve for Both Scenarios
For the equation |x| = 5, you solve for both x = 5 and x = -5. This gives you two possible solutions.
Step 4: Check Your Solutions
Always verify your solutions by substituting them back into the original equation. For example:
- If x = 5, then |5| = 5, which is correct.
- If x = -5, then |-5| = 5, which is also correct.
Example Problem
Solve the equation |2x – 3| = 7.
Solution:
Set up two equations:
- 2x – 3 = 7
- 2x – 3 = -7
Solve the first equation:
2x – 3 = 7
2x = 10
x = 5
Solve the second equation:
2x – 3 = -7
2x = -4
x = -2
Check the solutions:
- If x = 5, then |2(5) – 3| = |10 – 3| = |7| = 7, which is correct.
- If x = -2, then |2(-2) – 3| = |-4 – 3| = |-7| = 7, which is also correct.
Therefore, the solutions are x = 5 and x = -2.